Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.420=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Afghanistan	68% (51%-87%)	12,456	227
Albania	81% (35%-100%)	1,050	33
Algeria	30% (20%-42%)	8,857	623
Andorra	22% (11%-39%)	763	51
Argentina	32% (25%-42%)	13,920	500
Armenia	72% (52%-94%)	7,774	98
Australia	86% (53%-100%)	7,139	103
Austria	63% (37%-92%)	16,515	645
Azerbaijan	86% (63%-100%)	4,568	54
Bahamas	56% (15%-100%)	100	11
Bahrain	98% (89%-100%)	9,633	15
Bangladesh	66% (49%-89%)	38,292	544
Belarus	99% (92%-100%)	38,956	214
Belgium	24% (18%-30%)	57,592	9,364
Bolivia	24% (19%-32%)	7,768	280
Bosnia and Herzegovina	20% (11%-37%)	2,435	151
Brazil	19% (16%-23%)	411,821	25,598
Bulgaria	23% (17%-33%)	2,460	133
Burkina Faso	34% (17%-67%)	845	53
Cameroon	51% (29%-82%)	5,436	177
Canada	19% (15%-23%)	87,508	6,765
Chad	36% (19%-59%)	715	64
Chile	75% (58%-94%)	82,289	841
China	98% (98%-100%)	84,106	4,638
Colombia	38% (30%-48%)	24,104	803
Congo	40% (18%-79%)	571	19
Costa Rica	69% (27%-100%)	984	10
Cote dIvoire	88% (63%-100%)	2,556	31
Croatia	23% (12%-40%)	2,244	101
Cuba	56% (30%-90%)	1,974	82
Cyprus	85% (51%-100%)	939	17
Czechia	36% (27%-48%)	9,086	317
Democratic Republic of the Congo	60% (37%-91%)	2,659	68
Denmark	61% (42%-87%)	11,480	565
Dominican Republic	79% (63%-94%)	15,723	474
Ecuador	19% (11%-27%)	38,103	3,275
Egypt	37% (28%-46%)	19,666	816
El Salvador	62% (39%-92%)	2,109	39
Estonia	36% (22%-59%)	1,840	66
Finland	84% (45%-100%)	6,692	313
France	15% (11%-18%)	145,746	28,596
Gabon	92% (71%-100%)	2,319	14
Georgia	75% (36%-100%)	735	12
Germany	33% (26%-41%)	179,717	8,411
Ghana	97% (88%-100%)	7,303	34
Greece	23% (13%-34%)	2,903	173
Guatemala	48% (34%-68%)	4,145	68
Guernsey	46% (14%-97%)	252	13
Guinea	94% (77%-100%)	3,446	21
Guyana	52% (13%-99%)	139	11
Haiti	42% (20%-75%)	1,320	34
Honduras	34% (23%-48%)	4,640	194
Hungary	12% (8%-17%)	3,816	509
Iceland	87% (54%-100%)	1,805	10
India	39% (32%-46%)	158,333	4,531
Indonesia	23% (18%-29%)	23,851	1,473
Iran	53% (43%-63%)	141,591	7,564
Iraq	24% (16%-36%)	5,135	175
Ireland	35% (23%-52%)	24,803	1,631
Isle of Man	31% (9.1%-91%)	336	24
Israel	81% (59%-99%)	16,793	281
Italy	14% (11%-17%)	231,139	33,072
Japan	11% (7.8%-15%)	16,651	858
Jersey	14% (6.4%-35%)	308	29
Kazakhstan	98% (89%-100%)	9,576	37
Kenya	45% (21%-88%)	1,471	55
Kosovo	60% (34%-97%)	1,047	30
Kuwait	95% (82%-100%)	23,267	175
Kyrgyzstan	84% (54%-100%)	1,594	16
Latvia	52% (27%-91%)	1,057	23
Lebanon	76% (43%-100%)	1,161	26
Liberia	21% (8%-56%)	266	27
Lithuania	26% (12%-42%)	1,647	66
Luxembourg	48% (29%-69%)	4,001	110
Malaysia	96% (77%-100%)	7,619	115
Mali	17% (11%-25%)	1,116	70
Mauritius	62% (18%-100%)	334	10
Mexico	9.2% (7.5%-11%)	78,023	8,597
Moldova	30% (23%-39%)	7,537	274
Morocco	98% (83%-100%)	7,601	202
Netherlands	24% (18%-31%)	45,768	5,871
New Zealand	52% (25%-93%)	1,154	22
Niger	15% (7.7%-26%)	955	64
Nigeria	46% (33%-62%)	8,733	254
North Macedonia	23% (15%-33%)	2,040	119
Norway	73% (43%-98%)	8,383	235
Oman	97% (86%-100%)	8,373	38
Pakistan	65% (53%-79%)	61,227	1,260
Panama	56% (40%-74%)	11,728	315
Paraguay	91% (66%-100%)	884	11
Peru	36% (29%-43%)	135,905	3,983
Philippines	42% (30%-56%)	15,049	904
Poland	43% (33%-54%)	22,473	1,028
Portugal	30% (22%-39%)	31,292	1,356
Puerto Rico	64% (43%-90%)	3,397	129
Qatar	90% (50%-100%)	48,947	30
Romania	25% (19%-34%)	18,594	1,219
Russia	67% (55%-80%)	370,680	3,968
San Marino	81% (36%-100%)	667	42
Saudi Arabia	99% (93%-100%)	78,541	425
Senegal	83% (57%-100%)	3,253	39
Serbia	94% (72%-100%)	11,275	240
Sierra Leone	18% (11%-31%)	782	45
Singapore	91% (59%-100%)	32,876	23
Sint Maarten	12% (4.1%-32%)	77	15
Slovakia	70% (41%-99%)	1,515	28
Slovenia	19% (11%-34%)	1,471	107
Somalia	56% (31%-86%)	1,731	67
South Africa	34% (24%-44%)	25,937	552
South Korea	50% (22%-87%)	11,344	269
Sudan	25% (17%-35%)	4,146	184
Sweden	15% (11%-19%)	35,088	4,220
Switzerland	25% (19%-33%)	30,678	1,647
Tajikistan	86% (62%-100%)	3,100	46
Thailand	77% (49%-100%)	3,054	57
Togo	67% (30%-100%)	395	13
Tunisia	47% (17%-95%)	1,051	48
Turkey	77% (62%-90%)	158,762	4,397
Ukraine	44% (32%-59%)	21,584	644
United Arab Emirates	99% (93%-100%)	31,969	255
United Kingdom	24% (20%-28%)	267,240	37,460
United Republic of Tanzania	55% (28%-94%)	509	21
United States of America	34% (28%-40%)	1,699,933	100,442
Uruguay	43% (21%-81%)	803	22
Uzbekistan	94% (77%-100%)	3,333	14
Venezuela	90% (64%-100%)	1,245	11
Yemen	4.2% (2.7%-6.2%)	255	53

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

1 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim